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Theorem trlnle 30351
Description: The atom not under the fiducial co-atom  W is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
Hypotheses
Ref Expression
trlne.l  |-  .<_  =  ( le `  K )
trlne.a  |-  A  =  ( Atoms `  K )
trlne.h  |-  H  =  ( LHyp `  K
)
trlne.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlne.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnle  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )

Proof of Theorem trlnle
StepHypRef Expression
1 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  HL )
2 hlatl 29526 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
31, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  K  e.  AtLat )
4 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  P  e.  A )
5 trlne.l . . . . 5  |-  .<_  =  ( le `  K )
6 eqid 2380 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 trlne.a . . . . 5  |-  A  =  ( Atoms `  K )
85, 6, 7atnle0 29475 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  -.  P  .<_  ( 0. `  K ) )
93, 4, 8syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( 0. `  K ) )
10 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( K  e.  HL  /\  W  e.  H ) )
11 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
12 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  F  e.  T )
13 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( F `  P )  =  P )
14 trlne.h . . . . . 6  |-  H  =  ( LHyp `  K
)
15 trlne.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
16 trlne.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
175, 6, 7, 14, 15, 16trl0 30335 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =  P ) )  ->  ( R `  F )  =  ( 0. `  K ) )
1810, 11, 12, 13, 17syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( R `  F )  =  ( 0. `  K ) )
1918breq2d 4158 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  ( P  .<_  ( R `  F )  <->  P  .<_  ( 0. `  K ) ) )
209, 19mtbird 293 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =  P )  ->  -.  P  .<_  ( R `  F ) )
215, 7, 14, 15, 16trlne 30350 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )
2221adantr 452 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  =/=  ( R `  F
) )
23 simpl1l 1008 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  HL )
2423, 2syl 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  K  e.  AtLat )
25 simpl3l 1012 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  P  e.  A )
26 simpl1 960 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( K  e.  HL  /\  W  e.  H ) )
27 simpl3 962 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
28 simpl2 961 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  F  e.  T )
29 simpr 448 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( F `  P )  =/=  P )
305, 7, 14, 15, 16trlat 30334 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
3126, 27, 28, 29, 30syl112anc 1188 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( R `  F )  e.  A )
325, 7atncmp 29478 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  ( R `  F )  e.  A )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3324, 25, 31, 32syl3anc 1184 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  ( -.  P  .<_  ( R `
 F )  <->  P  =/=  ( R `  F ) ) )
3422, 33mpbird 224 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( F `
 P )  =/= 
P )  ->  -.  P  .<_  ( R `  F ) )
3520, 34pm2.61dane 2621 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  ( R `  F
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   class class class wbr 4146   ` cfv 5387   lecple 13456   0.cp0 14386   Atomscatm 29429   AtLatcal 29430   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   trLctrl 30323
This theorem is referenced by:  cdlemc3  30358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-undef 6472  df-riota 6478  df-map 6949  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324
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